Twenty-five families were asked how many TV sets they currently have in their ho

Question

Twenty-five families were asked how many TV sets they currently have in their home. Here are the data. 2, 3, 1, 2, 3, 0, 2, 4, 1, 2, 4, 3, 2, 1, 1, 3, 0, 2, 1, 1, 2, 3, 2, 5, 2 Find the mode, median, and mean. Show your steps when calculating the mean. Round the result to 2 decimal points. Calculate the range, variance, and standard deviation for inferential statistics. Show your steps when calculating the variance and standard deviation. Round the results to 2 decimal Assume that the number of TV sets a household has is normally distributed, approximately how many households would you expect to have either 1, 2, or 3 TV sets? For simplicity let's round the mean and the standard deviation to the nearest whole number.

Explanation / Answer

1. Mode refers to the most common score in the distribution. By looking into the given distribution, one can see 2 is the mode, as it has occurred 9 times.

Median is the point in a distribution of scores, above and below which exactly half of the cases falls. To compute median, sort the data in ascending order. Here, the distribution is odd in number (n=25), therefore, the middle most number is the median. Md=2 (13th number in the sorted list).

Mean is the arithmetic average of the scores. Therefore, xbar=sigma x/n, where, sigma x refer to sum of scores, and n is the number of scores.

xbar=(2+3+1+...+5+2)/25=2.08.

2. Range refers to the difference between highest score and lowest score, R=highest score-lowest score.

R=(5-0)=5

For computing variance and standard deviation fill the table as follows:

Note, mean, M=2.08

Variance, s^2=1/n-1 sigma (X-M)^2=1/25-1 *35.89=1.50

Standard deviation, s=sqrt s^2=1.22

3. Given, the data for TV sets a household ha sis normally distributed. Compute z score using following formula, z=(X-mu)/sigma, where, x is raw score, substitute the known values of mean and standard deviation to find number of households. Then find the subsequent area corresponding to the z score and multiply it with total number of households to find the required number of households having 1, 2 or 3 TV sets.

P(X=1)=P[Z=(1-2)/1]=P(Z=-1)=0.1587~15.87% households are expected to have 1 television set, which is around 4 households.

P(X=2)=P(Z=0)=0.5, therefore, around 13 households has 2 TV sets.

P(X=3)=P(Z=1)=0.8413, therefore, around 4 households has 3 TV sets.

X (X-M) (X-M)^2 5 2.92 8.53 4 1.92 3.67 4 1.92 3.67 3 0.92 0.85 3 0.92 0.85 3 0.92 0.85 3 0.92 0.85 3 0.92 0.85 2 -0.08 0.01 2 -0.08 0.01 2 -0.08 0.01 2 -0.08 0.01 2 -0.08 0.01 2 -0.08 0.01 2 -0.08 0.01 2 -0.08 0.01 2 -0.08 0.01 1 -1.08 1.17 1 -1.08 1.17 1 -1.08 1.17 1 -1.08 1.17 1 -1.08 1.17 1 -1.08 1.17 0 -2.08 4.33 0 -2.08 4.33 sigma X=52 sigma (X-M)=0 sigma (X-M)^2=35.89

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